Method of establishing anti-attack public key cryptogram

ABSTRACT

A method of establishing an anti-attack public key cryptogram includes (1) two parties select a braid group Bn with n≥7, and Bn=σ1, σ2, . . . , σn−1| σiσj=σjσi, |i−j|≥2, σiσi+1σi=σi+1σiσi+1, 1≤i≤n−2, (2) the two parties select two subgroups A and B in Bn generated from a1, a2, . . . , ak and b1, b2, . . . , bm respectively, (3) the first party selects an element x=x (a1, a2, . . . , ak)∈A as a first private key, and sends x−1b1x, x−1b2x, . . . , x−1bmx to the second party, (4) the second party selects an element y=y (b1, b2, . . . , bm)∈B as a second private key, and sends y−1a1y, y−1a2y, . . . , y−1aky to the first party, (5) the first party obtains KA=x−1x(y−1a1y, y−1a2y, . . . , y−1aky)=x−1y−1xy, (6) the second party obtains y−1y(x−1b1x, x−1b2x, . . . , x−1bkx)=y−1x−1yx, and calculates to obtain KB=(y−1x−1yx)−1=x−1y−1xy, thereby reaching a shared secret key K=KA=KB.

TECHNICAL FIELD

The present disclosure relates to the field of information security, andin particular to a method of establishing an anti-attack public keycryptogram.

BACKGROUND

Symmetric cryptography, such as AES, has proven to be a very efficientand secure method of transmitting confidential information. However, dueto the use of a symmetric secret key for encryption and decryption, bothparties during a confidential information transmission must establish ashared secret key through a secret key exchange protocol.

In a classic secret key sharing algorithm, as a practical calculationand security issue, its difficulty will be greatly reduced with theimprovement of computer performance. In particular, Shor proposed thefamous Shor quantum algorithm in 1997. The factorization of integers andthe calculation of discrete logarithms will be performed respectively inpolynomial time. This means that once quantum computer is achieved, thesecret key sharing protocol established based on RSA, ECC, ElGamalalgorithm, and the like will no longer be safe.

In Chinese patent number ZL201310382299.7, the structure of Mihailovasubgroups of a braid group with exponent n is disclosed. The membershipproblem of the subgroup is unsolvable. This serves as a core element inestablishing a new and highly secure cryptosystem.

SUMMARY

In order to solve the potential security problem based on the existingsecret key sharing protocol, an object of the present disclosure is toestablish a public key cryptographic method against various attacks bythe innovative introduction of unsolvability of a subgroup membershipproblem in a Mihailova subgroup of a braid group, and the conjugationproperty of the elements of the group.

The object of the present disclosure can be achieved by a method ofestablishing an anti-attack public key cryptogram, including thefollowing steps:

-   -   (1) two parties of a protocol select a braid group B_(n) with        exponent n≥7, and the braid group B_(n) is defined by the        following presentation:

B _(n)=

σ₁, σ₂, . . . , σ_(n−1)| σ_(i)σ_(j)=σ_(j)σ_(i), |i−j|≥2,σ_(i)σ_(i+1)σ_(i)=σ_(i+)1σ_(i)σ_(i+1), 1≤i≤n−2

,

-   -    each element of the braid group B_(n) being denoted by a word        on a set {σ₁, σ₂, . . . , σ_(n−1)} that represents the element,        possesses uniqueness and takes a normal form;    -   (2) the two parties of the protocol select two sets of elements        in B_(n), a₁, a₂, . . . , a_(k) and b₁, b₂, . . . , b_(m), to        generate two subgroups A and B of B_(n) respectively;    -   (3) a first party of the two parties selects an element x=x(a₁,        a₂, . . . , a_(k) )∈A as a first private key, and sends x⁻¹b₁x,        x⁻¹b₂x, . . . , x⁻¹b_(m)x to a second party of the two parties;    -   (4) the second party selects an element y=y(b₁, b₂, . . . ,        b_(m))∈B as a second private key, and sends y⁻¹a₁y, y⁻¹a₂y, . .        . , y⁻¹a_(k)y to the first party;    -   (5) after the first party receives the elements sent by the        second party, the first party replaces all a_(i) in x with        y⁻¹a_(i)y (i=1, 2, . . . , k) to obtain

K _(A) =x ⁻¹ x(y ⁻¹ a ₁ y, y ⁻¹ a ₂ y, . . . , y ⁻¹ a _(k) y)=x ⁻¹ y ⁻¹xy;

-   -   (6) after the second party receives the elements sent by the        first party, the second party replaces all b_(j) in y with        x⁻¹b_(j)x (j=1, 2, . . . , m) to obtain

y ⁻¹ y(x ⁻¹ b ₁ x, x ⁻¹ b ₂ x, . . . , x ⁻¹ b _(k) x)=y ⁻¹ x ⁻¹ yx,

-   -    and calculates to obtain

K _(B)=(y ⁻¹ x ⁻¹ yx)⁻¹ =x ⁻¹ y ⁻¹ xy;

-   -    since K_(A)=K_(B), the first party of the protocol and the        second party of the protocol reach a shared secret key        K=K_(A)=K_(B).

In a preferred embodiment, the braid group B_(n) is a Mihailova subgrouphaving an unsolvable subgroup membership, and both the subgroups A and Bare Mihailova subgroups.

In a preferred embodiment, the braid group B_(n) is a group defined bythe following presentation:

B _(n)=

σ₁, σ₂, . . . , σ_(n−1)| σ_(i)σ_(j)=σ_(j)σ_(i), |i−j|≥2,σ_(i)σ_(i+1)σ_(i)=σ_(i+1)σ_(i)σ_(i+1), 1≤i≤n−2

,

each element of the braid group B_(n) is denoted by a word on a set {σ₁,σ₂, . . . , σ_(n−1)} that represents the element, possesses uniquenessand takes a normal form;

when n≥7, B_(n) contains two subgroups isomorphic to F₂×F₂, i.e., twosubgroups isomorphic to the direct product of two free groups with arank of 2:

P=

σ₁ ², σ₂ ², σ₄ ², σ₅ ²

and

=

σ₂ ², σ₃ ², σ₅ ², σ₆ ²

;

then use a presentation that has a finite presentation group whose wordproblem is unsolvable and is generated by two elements so that subgroupA of P is a Mihailova subgroup, and subgroup B of Q is a Mihailovasubgroup;

56 generators of A are shown below, wherein i=1, and the 56 generatorsof A are obtained; and if i=2, 56 generators of B are obtained:

σ_(i) ²σ_(i+3) ², σ_(i+1) ²σ_(i+4) ², S_(ij), T_(ij), j=1, 2, . . . , 27

while 27 S_(ij) are:

-   S_(i1): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹²)⁻¹    σ_(i+1) ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i2): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰)⁻¹    σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i3): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸)⁻¹    σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i4): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶)⁻¹    σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i5): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ⁴)⁻¹ σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)    ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i6): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)¹ σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i7): (σ_(i) ²σ_(i+1) ⁴σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹²)⁻¹ σ_(i+1)    ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i8): (σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰)⁻¹ σ_(i+1)    ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i9): (σ_(i) ²σ_(i+1) ⁸σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸)⁻¹ σ_(i+1)    ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,10): (σ_(i) ²σ_(i+1) ¹⁰σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1)    ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶)⁻¹    σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,11): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁻¹    σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,12): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁻¹    σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,13): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹²)⁻¹    σ_(i+1) ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,14): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰)⁻¹    σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,15): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ⁸)⁻¹ σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,16): (σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i)    ⁴σ_(i+1) ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²⁰σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)    ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴σ_(i) ⁻⁴σ_(i+1)    ⁻²⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1) ⁻²⁰σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶-   S_(i,17): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i)    ⁴σ_(i+1) ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²⁰σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)    ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1)    ⁻²⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1) ⁻²⁰σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴-   S_(i,18): (σ_(i) ⁻⁴σ_(i+1) ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹²σ_(i) ⁴σ_(i+1) ⁻¹²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰σ_(i)    ⁻²σ_(i+1) ²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁻²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴ σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁻²σ_(i)    ⁻²σ_(i+1) ⁻¹⁸σ_(i) ² σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1)    ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²⁰(σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1))²)² σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ (σ_(i) ⁻⁴σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ²)⁻²σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ²σ_(i) ²σ_(i+1)    ²σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻¹⁸σ_(i) ²    σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1) ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²⁰(σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i)    ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²)³ σ_(i) ⁻⁴σ_(i+1)    ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹²σ_(i) ⁴σ_(i+1) ⁻¹²σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰-   S_(i,19): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1)    ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ² σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1)    ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ² σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i)    ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²-   S_(i,20): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)² σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁸σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1)    ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ² σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁴σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴    σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1)    ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁸ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)    ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)²σ_(i) ⁻⁴σ_(i+1)    ¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²-   S_(i,21): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)³ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁶σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1)    ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁶)²σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)²    σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1)    ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁶ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)    ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)³σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²-   S_(i,22): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁴ σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁸σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1)    ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁶)³σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)³    σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1)    ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁸ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)    ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁴σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²-   S_(i,23): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁵ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁰σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i)    ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶ σ_(i) ⁻⁴σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ²(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁴    σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)    ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i)    ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1)    ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁴ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i)    ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰ (σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁵σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²-   S_(i,24): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁶ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁰σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i)    ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸ σ_(i) ⁻⁴σ_(i+1)    ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁵    σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)    ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i)    ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1)    ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁵ σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ⁸σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i)    ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰ (σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²-   S_(i,25): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁷ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁶σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ² 94 _(i) ⁻²σ_(i+1) ⁸σ_(i)    ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸ σ_(i) ⁻⁴σ_(i+1)    ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰ (σ_(i)    ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)    ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1)    ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁶ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i)    ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i)    ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1)    ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁷σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²-   S_(i,26): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁸ (σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ²)³(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)    ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁷ σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)    ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁷ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶ 94 _(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ⁶(σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i)    ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²)³ (σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁸σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²-   S_(i,27): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁹ (σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ²)³(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)    ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁸ σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)    ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁸ σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁸(σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1)    ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²)³ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁹σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²    when all σ_(i) in each S_(ij) are replaced by σ_(i+3) and all    σ_(i+1)in each S_(ij) are replaced by σ_(i+4), 27 corresponding    T_(ij), j=1, 2, . . . , 27 are obtained.

In a preferred embodiment, the first private key x and the secondprivate key y are selected to be not less than 78 bits.

The shared secret key generated by the present disclosure is unsolvableby a third party. It serves as a core element in establishing a new andhighly secure cryptosystem. The security and equivalence of unsolvableproblem of the algorithm of the present disclosure can prove that it isimmune to all attacks. In addition, since the secret key sharing methodof the present disclosure uses unsolvable determination problem as asecurity guarantee, therefore the method is greatly secure boththeoretically and in actual application aspect. Compared with the priorart, the present disclosure has the following advantages:

1. It is theoretically proved that all attacks to the secret key sharingalgorithm of the present disclosure are not computable, and hence thesecret key sharing algorithm of the present disclosure can resist allknown attacks, including quantum computing attack.

2. Some private key selections are more secure due to the unsolvabilityof the Mihailova subgroup membership problem.

DETAILED DESCRIPTION

The secret key sharing protocol of the present disclosure againstquantum computing attack will be further described in detail below withreference to the embodiments.

1. Establish a Public Key Cryptographic Protocol Platform

Establish braid group B_(n) with exponent of n≥7, and two Mihailovasubgroups A and B with unsolvable subgroup membership problem. Inaddition, due to the demand of cryptogram and secret key generation,B_(n) must further satisfy the following conditions:

1) The word that represents the element of B_(n) on the set ofgenerators of B_(n) takes a computable normal form;

2) B_(n) is in exponential growth, i.e., the number of elements whoseword length is a positive integer n, B_(n) is confined to an exponentialfunction about n;

3) Multiplication and inversion of a group based on normal form iscomputable.

Therefore, the selected braid group B_(n) with exponent of n≥7 has theabove characteristics, and the group B_(n) is defined by the followingpresentation:

B _(n)=

σ₁, σ₂, . . . , σ_(n−1)| σ_(i)σ_(j)=σ_(j)σ_(i), |i−j|≥2,σ_(i)σ_(i+1)σ_(i)=σ_(i+1)σ_(i)σ_(i+1), 1≤i≤n−2

,

each element of the braid group B_(n) being denoted by a word on a set{σ₁, σ₂, . . . , σ_(n−1)} that represents the element, possessesuniqueness and takes a normal form.

When n≥7, B_(n) contains two subgroups isomorphic to F₂×F₂, i.e., twosubgroups isomorphic to the direct product of two free groups with arank of 2:

P=

σ₁ ², σ₂ ², σ₄ ², σ₅ ²

and

=

σ₂ ², σ₃ ², σ₅ ², σ₆ ²

;

then use a presentation that has a finite presentation group whose wordproblem is unsolvable and is generated by two elements so that subgroupA of P is a Mihailova subgroup, and subgroup B of

is a Mihailova subgroup;

56 generators of A are shown below, wherein i=1, and the 56 generatorsof A are obtained (if i=2, 56 generators of B are obtained):

σ_(i) ²σ_(i+3) ², σ_(i+1) ²σ_(i+4) ², S_(ij), T_(ij), j=1, 2, . . . , 27

while 27 S_(ij) are:

-   S_(i1): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹²)⁻¹    σ_(i+1) ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i2): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰)⁻¹    σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i3): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸)⁻¹    σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i4): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶)⁻¹    σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i5): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ⁴)⁻¹ σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)    ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i6): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)¹ σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i7): (σ_(i) ²σ_(i+1) ⁴σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹²)⁻¹ σ_(i+1)    ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i8): (σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰)⁻¹ σ_(i+1)    ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i9): (σ_(i) ²σ_(i+1) ⁸σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸)⁻¹ σ_(i+1)    ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,10): (σ_(i) ²σ_(i+1) ¹⁰σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1)    ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶)⁻¹    σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,11): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁻¹    σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,12): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁻¹    σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,13): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹²)⁻¹    σ_(i+1) ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,14): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰)⁻¹    σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,15): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ⁸)⁻¹ σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²-   S_(i,16): (σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i)    ⁴σ_(i+1) ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²⁰σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)    ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴σ_(i) ⁻⁴σ_(i+1)    ⁻²⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1) ⁻²⁰σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶-   S_(i,17): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i)    ⁴σ_(i+1) ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²⁰σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)    ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1)    ⁻²⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1) ⁻²⁰σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴-   S_(i,18): (σ_(i) ⁻⁴σ_(i+1) ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹²σ_(i) ⁴σ_(i+1) ⁻¹²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰σ_(i)    ⁻²σ_(i+1) ²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁻²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ⁴ σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ²σ_(i) ²σ_(i+1) ²σ_(i)    ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i)    ⁻²σ_(i+1) ²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁻²σ_(i)    ⁻²σ_(i+1) ⁻¹⁸σ_(i) ² σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1)    ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²⁰(σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1))²)² σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ (σ_(i) ⁻⁴σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ²)⁻²σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ²σ_(i) ²σ_(i+1)    ²σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻¹⁸σ_(i) ²    σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1) ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²⁰(σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i)    ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²)³ σ_(i) ⁻⁴σ_(i+1)    ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹²σ_(i) ⁴σ_(i+1) ⁻¹²σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰-   S_(i,19): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1)    ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ² σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1)    ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ² σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i)    ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²-   S_(i,20): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)² σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁸σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1)    ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ² σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁴σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴    σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1)    ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁸ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)    ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)²σ_(i) ⁻⁴σ_(i+1)    ¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²-   S_(i,21): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)³ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁶σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1)    ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁶)²σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)²    σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1)    ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁶ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)    ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)³σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²-   S_(i,22): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁴ σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁸σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1)    ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁶)³σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)³    σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1)    ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁸ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)    ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁴σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²-   S_(i,23): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁵ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁰σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i)    ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶ σ_(i) ⁻⁴σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ²(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁴    σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)    ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i)    ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1)    ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁴ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i)    ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰ (σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁵σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²-   S_(i,24): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁶ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ¹⁰σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i)    ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸ σ_(i) ⁻⁴σ_(i+1)    ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁵    σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)    ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i)    ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1)    ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁵ σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ⁸σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i)    ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰ (σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²-   S_(i,25): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁷ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁶σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ² 94 _(i) ⁻²σ_(i+1) ⁸σ_(i)    ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸ σ_(i) ⁻⁴σ_(i+1)    ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)    ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰ (σ_(i)    ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)    ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1)    ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁶ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i)    ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i)    ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1)    ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁸)⁷σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²-   S_(i,26): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁸ (σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ²)³(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)    ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁷ σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)    ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁷ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶ 94 _(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ⁶(σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i)    ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²)³ (σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁸σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)    ²σ_(i+1) ²-   S_(i,27): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)    ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁹ (σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ²)³(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)    ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁸ σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)    ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1)    ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)    ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁸ σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1)    ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)    ⁸(σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1)    ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²)³ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)    ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)    ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁹σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)    ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²

2. Establish Protocol of a Secret Key Sharing System

In the present embodiment, for example, the two parties of the protocolare Alice and Bob respectively.

-   -   (1) Alice and Bob select a braid group B_(n) with exponent n≥7,        and the braid group B_(n) is defined by the following        presentation:

B _(n)=

σ₁, σ₂, . . . , σ_(n−1)| σ_(i)σ_(j)=σ_(j)σ_(i) , |i−j|≥2,σ_(i)σ_(i+1)σ_(i)=σ_(i+1)σ_(i)σ_(i+1), 1≤i≤n−2

,

-   -    each element of the braid group B_(n) being denoted by a word        on a set {σ₁, σ₂, . . . , σ_(n−1)} that represents the element,        possesses uniqueness and takes a normal form;    -   (2) Alice and Bob respectively select two sets of elements, a₁,        a₂, . . . , a_(k)∈A and b₁, b₂, . . . , b_(m)∈B, of two        Mihailova subgroups A and B in B_(n) respectively;    -   (3) Alice selects an element x=x (a₁, a₂, . . . , a_(k))∈A as a        first private key, and sends x⁻¹b₁x, x⁻¹b₂x, . . . , x⁻¹b_(m)x        to Bob;    -   (4) Bob selects an element y=y (b₁, b₂, . . . , b_(m))∈B as a        second private key, and sends y⁻¹a₁y, y⁻¹a₂y, . . . , y⁻¹a_(k)y        to Alice;    -   (5) Alice replaces all a_(i) in x with y⁻¹a_(i)y (i=1, 2, . . .        , k) to obtain

K _(A) =x ⁻¹ x(y ⁻¹ a ₁ y, y ⁻¹ a ₂ y, . . . , y ⁻¹ a _(k) y)=x ⁻¹ y ⁻¹xy;

-   -   (6) Bob replaces all b_(j) in y with x⁻¹b_(j)x (j=1, 2, . . .        , m) to obtain

y ⁻¹ y(x ⁻¹ b ₁ x, x ⁻¹ b ₂ x, . . . , x ⁻¹ b _(k) x)=y ⁻¹ x ⁻¹ yx,

-   -    and calculates to obtain

K _(B)=(y ⁻¹ x ⁻¹ yx)⁻¹ =x ⁻¹ y ⁻¹ xy;

-   -    since K_(A)=K_(B), Alice and Bob reach a shared secret key        K=K_(A)=K_(B).

Security Analysis:

First, the definitions of the two determination problems in the groupare given.

Subgroup Membership Problem or Generalized Word Problem (GWP):

Given a subgroup H whose generator set is X in group G, determinewhether any element g in G can be represented by a word on X, i.e.,whether g is an element in H or not.

Conjugacy Search Problem (CSP):

Given that g and h are two elements in group G, and that there is anelement c in G so that h=c⁻¹gc, determine whether there is element c′ inH so that h=c′⁻¹gc′.

In the secret key sharing protocol, attacker Eve can obtain thefollowing information through public information and interactive processof Alice and Bob: braid group B_(n) with exponent n≥7, two sets ofgenerated elements a₁, a₂, . . . , a_(k)∈A and b₁, b₂, . . . , b_(m)∈Bof two Mihailova subgroups A and B of B_(n), and elements x⁻¹b₁x,x⁻¹b₂x, . . . , x⁻¹b_(m)x and y⁻¹a₁y, y⁻¹a₂y, . . . , y⁻¹a_(k)y inB_(n).

Eve only knows the normal form of the words that represent theseelements of x⁻¹b₂x, . . . , x⁻¹b_(m)x and y⁻¹a₁y, y⁻¹a₂y, . . . ,y⁻¹a_(k)y. However, Eve does not know the corresponding decomposedexpressions.

If Eve can get x′∈B_(n) and y′∈B_(n) by solving the CSP problem so thatx′⁻¹b_(i)x′=x⁻¹b_(i)x and y′⁻¹a_(j)y′=y⁻¹a_(j)y, i=1, 2, . . . , m, j=1,2, . . . , k, however, Eve cannot guarantee x′=x and y′=y. Assumingx′=c_(a)x, y′=c_(b)y, then obtain (c_(a)x)⁻¹b_(i)c_(a)x=x⁻¹c_(a) ⁻¹b_(i)c_(a)x=x⁻¹b_(i)x from x′⁻¹b_(i)x′=x⁻¹b_(i)x and y′⁻¹a_(j)y′=y⁻¹a_(j)y sothat c_(a) ⁻¹b_(i)c_(a)=b_(i). That is, c_(a) and b_(i) can beexchanged, i=1, 2, . . . , m. Similarly, c_(b) and a_(j) can beexchanged, j=1, 2, . . . , k. In addition, Eve must require(c_(a)x)⁻¹(c_(b)y)⁻¹c_(a)xc_(b)y=x⁻¹c_(a) ⁻¹y⁻¹c_(b)⁻¹c_(a)xc_(b)y=x⁻¹y⁻¹c_(a) ⁻¹c_(b) ⁻¹c_(a)c_(b)xy=x⁻¹y⁻¹xy=K, and thenthere must be c_(b)c_(a)=c_(a)c_(b). Therefore, only when x′∈A and y′∈B,then there are c_(a)∈A and c_(b)∈B. Since c_(a) is exchangeable with allb_(i), c_(a) centralizes subgroups B. So when c_(b)c_(a)=c_(a)c_(b), Evecan get the correct shared secret key K. That is, Eve must know that x′and y′ she obtained by solving the CSP problem are the elements ofsubgroup A and the elements of subgroup B, respectively. Thus, she mustsolve the membership problem of the elements of Mihailova subgroup A ofx′ and y′ and subgroup B. However, this problem is unsolvable.Therefore, the secret key sharing protocol is secure. Eve cannot carryout an attack even with quantum computing system.

Choosing of a Parameter:

In a preferred embodiment, the braid group B_(n) has exponent n≥7. Thefirst private key x and the second private key y of the protocol areselected to be not less than 78 bits.

The foregoing describes a method for establishing anti-attack public keycryptogram of the present disclosure to help to understand the presentdisclosure. However, the implementation manners of the presentdisclosure are not limited by the foregoing embodiments. Any variation,modification, replacement, combination, and simplification made withoutdeparting from the principle of the present disclosure shall be anequivalent replacement manner and fall within the scope of protection ofthe present disclosure.

1. A method of establishing an anti-attack public key cryptogram,comprising the following steps: (1) two parties of a protocol select abraid group B_(n) with exponent n≥7, and the braid group B_(n) isdefined by the following presentation:B _(n)=

σ₁, σ₂, . . . , σ_(n−1)| σ_(i)σ_(j)=σ_(j)σ_(i) , |i−j|≥2,σ_(i)σ_(i+1)σ_(i)=σ_(i+1)σ_(i)σ_(i+1), 1≤≤n−2

,  each element of the braid group B_(n) being denoted by a word on aset {σ₁, σ₂, . . . , σ_(n−1)} that represents the element, possessesuniqueness and takes a normal form; (2) the two parties of the protocolselect two sets of elements in B_(n), a₁, a₂, . . . , a_(k) and b₁, b₂,. . . b_(m), to generate two subgroups A and B of B_(n) respectively;(3) a first party of the two parties selects an element x=x (a₁, a₂, . .. , a_(k))∈A as a first private key, and sends x⁻¹b₁x, x⁻¹b₂x, . . . ,x⁻¹b_(m)x to a second party of the two parties; (4) the second partyselects an element y=y (b₁, b₂, . . . , b_(m))∈B as a second privatekey, and sends y⁻¹a₁y, y⁻¹a₂y, . . . , y⁻¹a_(k)y to the first party; (5)after the first party receives the elements sent by the second party,the first party replaces all a_(i) in x with y⁻¹a_(i)y (i=1, 2, . . . ,k) to obtainK _(A) =x ⁻¹ x(y ⁻¹ a ₁ y, y ⁻¹ a ₂ y, . . . , y ⁻¹ a _(k) y)=x ⁻¹ y ⁻¹xy; (6) after the second party receives the elements sent by the firstparty, the second party replaces all b_(j) in y with x⁻¹b_(j)x (j=1, 2,. . . , m) to obtainy ⁻¹ y(x ⁻¹ b ₁ x, x ⁻¹ b ₂ x, . . . , x ⁻¹ b _(k) x)=y ⁻¹ x ⁻¹ yx,  andcalculates to obtainK _(B)=(y ⁻¹ x ⁻¹ yx)⁻¹ =x ⁻¹ y ⁻¹ xy;  since K_(A)=K_(B), the firstparty of the protocol and the second party of the protocol reach ashared secret key K=K_(A)=K_(B).
 2. The method of establishing ananti-attack public key cryptogram as claimed in claim 1, wherein thebraid group B_(n) is a Mihailova subgroup having an unsolvable subgroupmembership, and both the subgroups A and B are Mihailova subgroups. 3.The method of establishing an anti-attack public key cryptogram asclaimed in claim 1, wherein the braid group B_(n) is a group defined bythe following presentation:B _(n)=

σ₁, σ₂, . . . , σ_(n−1)| σ_(i)σ_(j)=σ_(j)σ_(i) , |i−j|≥2,σ_(i)σ_(i+1)σ_(i)=σ_(i+1)σ_(i)σ_(i+1), 1≤i≤n−2

, each element of the braid group B_(n) is denoted by a word on a set{σ₁, σ₂, . . . , σ_(n−1)} that represents the element, possessesuniqueness and takes a normal form; when n≥7, B_(n) contains twosubgroups isomorphic to F₂×F₂, i.e., two subgroups isomorphic to thedirect product of two free groups with a rank of 2:P=

σ₁ ², σ₂ ², σ₄ ², σ₅ ²

and

=

σ₂ ², σ₃ ², σ₅ ², σ₆ ²

; then use a presentation that has a finite presentation group whoseword problem is unsolvable and is generated by two elements so thatsubgroup A of P is a Mihailova subgroup, and subgroup B of

is a Mihailova subgroup; 56 generators of A are shown below, whereini=1, and the 56 generators of A are obtained; and if i=2, 56 generatorsof B are obtained:σ_(i) ²σ_(i+3) ², σ_(i+1) ²σ_(i+4) ², S_(ij), T_(ij), j=1, 2, . . . , 27while 27 S_(ij) are: S_(i1): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i)⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹²)⁻¹ σ_(i+1) ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i)⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ² S_(i2): (σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴σ_(i)⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰)⁻¹ σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i)⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ² S_(i3): (σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸)⁻¹ σ_(i+1) ⁻⁸σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1)⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ² S_(i4): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1)⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)⁶)⁻¹ σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁹σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1)⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ² S_(i5): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰σ_(i) ⁻⁴σ_(i+1)⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ⁴)⁻¹ σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹⁴)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i)⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ² S_(i6): (σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²(σ_(i)⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁻¹σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ² S_(i7): (σ_(i)²σ_(i+1) ⁴σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ⁴σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1)⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹²)⁻¹ σ_(i+1) ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1)⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ² S_(i8): (σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1)⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰)⁻¹ σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1)⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ² S_(i9): (σ_(i) ²σ_(i+1) ⁸σ_(i) ⁻²σ_(i+1)⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²(σ_(i) ⁻⁴σ_(i+1)⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸)⁻¹ σ_(i+1)⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ² S_(i,10): (σ_(i) ²σ_(i+1)¹⁰σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)²(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁹ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ⁶)⁻¹ σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ² S_(i,11):(σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁻¹ σ_(i+1)⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ² S_(i,12): (σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)⁴σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1)⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁻¹ σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1)⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ² S_(i,13): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1)⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹²)⁻¹ σ_(i+1) ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1)⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ² S_(i,14): (σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1)⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)¹⁰)⁻¹ σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1)⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ² S_(i,15):(σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁰σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸)⁻¹ σ_(i+1)⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ² S_(i,16): (σ_(i+1) ⁻⁶σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1) ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ²⁰σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴σ_(i) ⁻⁴σ_(i+1)⁻²⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1) ⁻²⁰σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ⁶ S_(i,17): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)²⁰σ_(i) ⁴σ_(i+1) ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²⁰σ_(i) ⁻⁴σ_(i+1)⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻²⁰σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1) ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴S_(i,18): (σ_(i) ⁻⁴σ_(i+1) ⁻¹²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹²σ_(i)⁴σ_(i+1) ⁻¹²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰σ_(i) ⁻²σ_(i+1) ²σ_(i)²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i)²σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ²σ_(i)²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1) ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ²⁰(σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i)⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1))²)² σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹(σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1)⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²)⁻²σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ²σ_(i)²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²⁰σ_(i) ⁴σ_(i+1) ⁻²⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ²⁰(σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i)⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²)³ σ_(i) ⁻⁴σ_(i+1) ⁻¹²σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹²σ_(i) ⁴σ_(i+1) ⁻¹²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹⁰ S_(i,19): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹⁴ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i)⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ² σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴σ_(i) ⁻⁴σ_(i+1)⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ² σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1)⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁶ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ² S_(i,20):(σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)²σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1)⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹⁴σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴ σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ²σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i)⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹⁶)²σ_(i) ⁻⁴σ_(i+1) ¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ² S_(i,21): (σ_(i+1)⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)³ σ_(i)⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴ (σ_(i)⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)²σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)² σ_(i) ⁻⁴σ_(i+1)⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ⁴σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶ (σ_(i) ⁻⁴σ_(i+1)⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁶)³σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ² S_(i,22):(σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁴σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1)⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)³σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)³ σ_(i)⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸ (σ_(i)⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁴σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²S_(i,23): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1)⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)¹⁴)⁵ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1)⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1)⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)¹⁶)⁴ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i)⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹⁴)⁴ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i)⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹⁰ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁵σ_(i) ⁻⁴σ_(i+1)⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ² S_(i,24): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁴)⁶ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰σ_(i) ⁻⁴σ_(i+1)⁻⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ⁸ σ_(i) ⁻⁴σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)⁴σ_(i) ⁴σ_(i+1) ⁻⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁴(σ_(i) ⁻⁴σ_(i+1)⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁶)⁵ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i)⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁵ σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸σ_(i)⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁶σ_(i)⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ² S_(i,25): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i)⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁷ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i)⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ² 94 _(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1) ⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i)⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁶σ_(i)⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1)⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1)¹⁴)⁶ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1)⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ⁶σ_(i) ⁴σ_(i+1)⁻⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶σ_(i) ⁻⁴σ_(i+1) ⁻¹⁰σ_(i) ²σ_(i+1)²σ_(i) ⁻²σ_(i+1) ¹⁰σ_(i) ⁴σ_(i+1) ⁻¹⁰σ_(i) ⁻¹σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁰(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1)⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁷σ_(i) ⁻⁴σ_(i+1) ⁻¹⁸σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ² S_(i,26): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i)⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁴σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹⁴)⁸ (σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i)⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²)³(σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i)²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i)²σ_(i+1) ¹⁶)⁷ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i)⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i) ⁻⁴σ_(i+1)⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁴)⁷ σ_(i) ⁻⁴σ_(i+1) ⁻⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)⁶σ_(i) ⁴σ_(i+1) ⁻⁶ 94 _(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁶(σ_(i) ⁻⁴σ_(i+1)⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ²)³ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁸σ_(i) ⁻⁴σ_(i+1)⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ² S_(i,27): (σ_(i+1) ⁻⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁸(σ_(i) ⁻⁴σ_(i+1)⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁴)⁹ (σ_(i) ⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)²σ_(i) ⁴σ_(i+1) ⁻²σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²)³(σ_(i) ⁻⁴σ_(i+1)⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1)⁻²σ_(i) ²σ_(i+1) ¹⁶)⁸ σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1)¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²)⁻¹ σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴(σ_(i)⁻⁴σ_(i+1) ⁻¹⁴σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁴σ_(i) ⁴σ_(i+1) ⁻¹⁴σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁴)⁸ σ_(i) ⁻⁴σ_(i+1) ⁻⁸σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ⁸σ_(i) ⁴σ_(i+1) ⁻⁸σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ⁸(σ_(i)⁻⁴σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ²σ_(i) ⁴σ_(i+1) ⁻²σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ²)³ (σ_(i) ⁻⁴σ_(i+1) ⁻¹⁶σ_(i) ²σ_(i+1) ²σ_(i)⁻²σ_(i+1) ¹⁶σ_(i) ⁴σ_(i+1) ⁻¹⁶σ_(i) ⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ¹⁶)⁹σ_(i)⁻⁴σ_(i+1) ⁻¹⁸σ_(i) ²σ_(i+1) ²σ_(i) ⁻²σ_(i+1) ¹⁸σ_(i) ⁴σ_(i+1) ⁻¹⁸σ_(i)⁻²σ_(i+1) ⁻²σ_(i) ²σ_(i+1) ² when all σ_(i) in each S_(ij) are replacedby σ_(i+3) and all σ_(i+1) in each S_(ij) are replaced by σ_(i+4), 27corresponding T_(ij), j=1, 2, . . . , 27 are obtained.
 4. The method ofestablishing an anti-attack public key cryptogram as claimed in claim 1,wherein the first private key x and the second private key y areselected to be not less than 78 bits.